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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 C - Spring 2007 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 10, 2007 \hfil \pp
\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hrulefill
\hspace{0.5in}
 Section: \hrulefill
\vfil
\begin{center}
{\large
\begin{tabular}{||c|c|r||} \hline 
 1 & 10 &\hspace{10mm} \hfil\\ \hline 
 2 & 10 &       \\ \hline
 3 & 10 &      \\ \hline
4 &  10 &    \\ \hline
5 & 10 & \\ \hline
Total & 50 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all questions. 
\item You may use a scientific, non-graphing calculator during this
examination.  
Other electronic devices are not allowed, and should be
turned off for the duration of the exam.
\item If you use a trial-and-error or guess-and-check method,
or read a numerical solution from a graph on your calculator,
when an algebraic method is available, you will not receive full credit.
\item You may use one hand-written 8.5 by 11 inch page of notes.
\item Show all work for full credit.  
\item You have 50 minutes to complete the exam. 
\end{itemize}
\vfil
.

\newp

\begin{enumerate}

\item Consider the curve defined by the vector equation 
\[
\overrightarrow r(t) = \langle 4t, 5t^3, 2t^2 \rangle
\]
\begin{enumerate}
\item Find the unit tangent vector $\overrightarrow T(t)$ at the 
point where $t=1$.

\vfil

\item Find the parametric equations of the tangent line the curve at the point where
$t=1$.

\end{enumerate}

\newp


\item Does the curve defined by the polar equation 
\[
r = \sec \theta + \tan \theta
\]
intersect the vertical line $x=2$? Explain.

\newp

\item Suppose a particle is moving in 3-dimensional space so that its
position vector is 
\[
\overrightarrow r(t) = \langle t,t^2,\frac{1}{t} \rangle.
\]

\begin{enumerate}
\item Find the tangential component of the particle's acceleration vector
at time $t=1$.
\vfil
\item Find all values of $t$ at which the particle's velocity
vector is orthogonal to the particle's acceleration vector.
\end{enumerate}

\newp

\item Consider the curve in the $xy$-plane defined by the position vector function
\[
\overrightarrow r(t) = \langle t^2-3t, t^2+2t \rangle
\]

Find the $t$-value of the point of maximum curvature on this curve.

\newp


\item Let $\dsps f(x,y) = x e^y - \ln(x+y)$.

\begin{enumerate}

\item Sketch the domain of $f$.

\vfil
\item Find $f_{xy}(x,y)$.

\end{enumerate}


\end{enumerate}

\end{document}


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